Exceptional curves on smooth rational surfaces with $-K$ not nef and of self-intersection zero

A $(-n)$-curve is a smooth rational curve of self-intersection $-n$, where $n$ is a positive integer. In 1998 Hirschowitz asked whether a smooth rational surface $X$ defined over the field of complex numbers, having an anti-canonical divisor not nef and of self-intersection zero, has $(-2)$-curves. In this paper we prove that for such a surface $X$, the set of $(-1)$-curves on $X$ is finite but non-empty, and that $X$ may have no $(-2)$-curves. Related facts are also considered.